SUNY Geneseo Department of Mathematics

Scalar Line Integrals

Thursday, April 23

Math 223 01
Spring 2020
Prof. Doug Baldwin

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Previous Lecture

Questions?

Scalar Line Integral Examples

Monday we saw the definition of a scalar line integral as a Riemann sum of a function’s values along an arbitrary curve times the lengths of short segments of that curve. We also saw a calculationally more practical formula that says the line integral of function f along curve r(t) between t = a and t = b is the regular integral from a to b of f(r(t)) ||r’(t)||.

The Idea

Integrate f(x,y) = x + √y along the curve r(t) = ⟨ t, t2 ⟩, 1 ≤ t ≤ 2.

A good starting place is to find r’(t) and its magnitude:

Parametric form for path, its derivative, and magnitude

Then you can plug the magnitude into the calculational formula for the line integral, and integrate as a single-variable integral:

Evaluating integral from 1 to 2 of f of r of t times magnitude of r prime

A More Complicated Path

Integrate g(x,y) = x2 + y2 around the path that follows a half circle of radius π from (π,0), above the x axis to (-π,0), then along the x axis back to (π,0):

Semicircle arching over the origin, with its flat side on the x axis

Here we need to treat the path as a piecewise function. We can use a theorem about line integrals that says a line integral over a series of contiguous but non-overlapping paths can be evaluated as the sum of the integrals over the individual paths.

Integral along path C equals integral along C 1 plus integral along C 2

In this case the arc over the top is one subpath and the straight line along the x axis is the another.

Parametric forms and their derivatives for a half circle and horizontal line segment

With the subpaths identified and parameterized, plug each into its own integral. Evaluating those integrals is simpler than it might look:

Evaluate sum of integrals along each part of path

Problem Set

On line integrals and vector fields. We’ve done some of this, but not yet all.

See the handout for details.

Next

Line integrals become more interesting, and more powerfully useful (e.g., in modeling force and work in physics), if the functions you are integrating are themselves vector-valued. Vector valued functions of points in space are called “vector fields.” So let’s learn something about vector fields.

Please read...

in section 15.1 of the textbook.

But do not read “Gradient Fields (Conservative Fields)” or beyond (even though it looks like it’s part of “Vector Fields in R3”).

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